For a small molecule with a diffusion coefficient of 10^-6 cm2/s, how far would it diffuse after flowing over 1 cm at a flow rate of 10 cm/s?
To determine how far the small molecule would diffuse after flowing over 1 cm at a flow rate of 10 cm/s, we can use Fick's First Law of Diffusion.
Fick's First Law of Diffusion states:
J = -D * (dC/dx),
where:
- J is the diffusion flux (amount of substance diffusing per unit area, per unit time)
- D is the diffusion coefficient of the substance
- dC/dx is the concentration gradient (change in concentration per unit distance)
In this case, we know:
- D = 10^-6 cm^2/s (given diffusion coefficient)
- x = 1 cm (distance traveled)
First, let's determine the concentration gradient (dC/dx):
Since the concentration gradient is not provided in the question, we'll assume it to be constant (which is commonly assumed for flow systems).
Therefore, dC/dx = C/x,
where C is the concentration.
Now, let's substitute the given values into Fick's First Law of Diffusion:
J = -D * (C/x).
The diffusion flux (J) represents the amount of substance diffusing per unit area, per unit time. In this case, since we're looking for the distance the molecule would diffuse, we can rearrange the equation to solve for J:
J = -D * (C/x) [divide both sides by -D]
J / -D = C/x [multiply both sides by x]
Jx / -D = C.
Now, let's substitute the known values into the equation:
J = 10 cm/s (given flow rate)
x = 1 cm (given distance)
C = Jx / -D
C = (10 cm/s) * (1 cm) / (10^-6 cm^2/s)
C = 10^-4 mol/cm^3.
Now that we have the concentration, we can substitute it into the equation for the concentration gradient (dC/dx) we assumed earlier:
dC/dx = C/x
dC/dx = (10^-4 mol/cm^3) / (1 cm)
dC/dx = 10^-4 mol/cm^4.
Now we can substitute both the diffusion coefficient (D) and the concentration gradient (dC/dx) into Fick's First Law of Diffusion:
J = -D * (dC/dx)
J = -(10^-6 cm^2/s) * (10^-4 mol/cm^4)
J = -10^-10 mol / (cm^3*s).
Now, we want to know how far the molecule would diffuse after flowing over 1 cm. Therefore, we need to determine how much of the molecule has diffused:
Amount of substance diffused (Δn) = J * A * t,
where:
- Δn is the amount of substance diffused
- J is the diffusion flux
- A is the surface area
- t is the time.
Since the molecule flows over 1 cm at a flow rate of 10 cm/s, the time (t) it takes to flow over 1 cm is t = 1 cm / 10 cm/s = 0.1 s.
To calculate the diffusion distance, we need to consider the surface area (A) and the time (t):
We'll assume a cylinder shape with cross-sectional area (A) of 1 cm^2.
Δn = J * A * t = (-10^-10 mol / (cm^3 * s)) * (1 cm^2) * (0.1 s)
Δn = -(10^-11 mol).
Now, assuming the molecule diffuses uniformly in all directions within the flowing fluid, the diffusion distance (d) can be calculated using the equation:
d = sqrt((6 * Δn) / (π * N * d0^2)),
where:
- d is the diffusion distance
- Δn is the amount of substance diffused
- N is Avogadro's number (6.022 x 10^23 mol^-1)
- d0 is the diameter of the molecule.
Since we do not have the diameter of the molecule in the question, we cannot calculate the diffusion distance accurately without that information.
Therefore, without knowing the diameter of the molecule, we cannot determine the exact diffusion distance.